** 1.CONSTRUCTION OF TRIANGLES**

**Any one of the following sets of measurements are required to construct a triangle**

• Length of the three sides

• Two sides and the included angle

• Two angles and the included side

• Length of the hypotenuse and one side in case of a right-angled triangle.**Construction of a triangle when measurements of its three sides are given**Construct ΔABC, when AB = 6 cm, BC = 7 cm and CA = 9 cm.

**Steps of construction:**

Step 1: Draw line segment BC = 7 cm.

Step 2: Draw an arc with B as the centre and the radius equal to 6 cm.

Step 3: Draw an arc with C as the centre and the radius equal to 9 cm.

Step 4: Name the point of intersection of these two arcs as A.

Step 5: Join points A and B, and points A and C.

Triangle ABC is the required triangle.**Construction of a triangle when measurements of two sides and the included angle are given**Construct ΔPQR, when PQ = 4 cm, QR = 6 cm and ∠PQR = 60°.

**Steps of construction:**

Step 1: Draw line segment QR = 6 cm.

Step 2: Construct an angle of 60° at point Q.

Step 3: Draw an arc on the ray QX with Q as the centre and the radius equal to 4 cm.

Step 4: Name the point where the arc cuts ray QX, as P.

Step 5: Join points P and R.

Triangle PQR is the required triangle.**Construction of a triangle, when two angles and the included side are given**Construct ΔXYZ, when ∠ZXY = 40°, ∠XYZ = 95° and the included side XY = 8 cm.

**Steps of construction:**

Step 1: Draw line segment XY = 8 cm.

Step 2: Construct an angle of 40° at X with XY.

Step 3: Construct another angle of 95° at Y with YX.

Step 4: Name the point of intersection of the two rays as Z.

Triangle XYZ is the required triangle.**Construction of a right-angled triangle, when the length of one side and the hypotenuse are given**Construct a right-angled triangle LMN, with hypotenuse LN = 8 cm and side MN = 5 cm.

**Steps of construction:**

Step 1: Draw line ‘*l*‘.

Step 2: Mark a point on ‘*l*‘ and name it M.

Step 3: Draw a line segment MN = 5 cm on ‘*l*‘ .

Step 4: Construct a right angle XMN at M.

Step 5: Draw an arc with N as the centre and radius equal to 8 cm, such that it intersects MX.

Step 6: Mark the point of intersection as L.

Step 7: Join points L and N.

Triangle LMN is the required triangle.

**2. CONSTRUCTION OF PARALLEL LINES**

Two lines in a plane that never meet each other at any point are said to be parallel to each other.

Any line intersecting a pair of parallel lines is called a transversal.

**Properties of angles formed by parallel lines and transversal**

• All pairs of alternate interior angles are equal.

• All pairs of corresponding angles are equal.

• All pairs of alternate exterior angles are equal.

• The interior angles formed on the same side of the transversal are supplementary (the sum of their measures is 180°).**Construction of a parallel line using the alternate interior angle property**

Step 1: Draw line ‘l’ and point A outside it.

Step 2: Mark point B on line ‘l’.

Step 3: Draw line ‘n’ joining point A and point B.

Step 4: Draw an arc with B as the centre, such that it intersects line ‘l’ at D and line ‘n’ at E.

Step 5: Draw another arc with the same radius and A as the centre, such that it intersects line ‘n’ at F. Ensure that arc drawn from A cuts the line ‘n’ between A and B.

Step 6: Draw another arc with F as the centre and distance DE as the radius.

Step 7: Mark the point of intersection of this arc and the previous arc as G.

Step 8: Draw line ‘m’ passing through points A and G.

Line ‘m’ is the required parallel line.**Verification of the construction**

If the pair of alternate interior angles are equal in measure, then line ‘m’ is parallel to line ‘*l*‘.

**Construction of a parallel line using the corresponding angle property**

Step 1: Draw line ‘l’ and point P outside it.

Step 2: Mark point Q on line ‘l’.

Step 3: Draw line ‘n’ joining point P and point Q.

Step 4: Draw an arc with Q as the centre, such that it intersects line ‘l’ at R and line ‘n’ at S.

Step 5: Draw another arc with the same radius and P as the centre, such that it intersects line ‘n’ at X. Ensure that arc drawn from P cuts the line ‘n’ outside QP.

Step 6: Draw another arc with X as the centre and distance RS as the radius, such that it intersects the previous at Y.

Step 7: Draw line ‘m’ passing through points P and Y.

Line ‘m’ is the required parallel line.**Verification of the construction**

If the pair of corresponding angles are equal in measure, then line ‘m’ is parallel to line ‘*l*‘.